Explore the world of polynomials in mathematics, from their definition as expressions involving variables and coefficients to their representation in MATLAB. Learn about finding the value, plotting, and roots of polynomials, with practical examples and solutions provided.
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Polynomials: In mathematics, a polynomial is an expression consisting of variables and coefficients, involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single variable x is x^2 4x + 7. An example in three variables is x^3 + 2xyz^2 yz + 1. Polynomials appear in a wide variety of areas of mathematics and science. Polynomials are functions that have the form: that The coefficients an, an-1 ... , a1, a0are real numbers, and n which is a nonnegative integer, is the degree, or order, of the polynomial. Examples of polynomials are:
In MATLAB, polynomials are represented by a row vector in which the elements are the coefficients an, an-1, ... , a1, a0. For example Value of a Polynomial The value of a polynomial at a point x can be calculated with the function polyval that has the form:
x can also be a vector or a matrix. In such a case the polynomial is calculated for each element (element-by-element), and the answer is a vector, or a matrix, with the corresponding values of the polynomial EX1/ Solution The problem is solved in the Command Window. (a) The coefficients of the polynomials are assigned to vector p. The function polyval is then used to calculate the value at x = 9.
Roots of a Polynomial The roots of a polynomial are the values of the argument for which the value of the polynomial is equal to zero. MATLAB has a function, called roots, that determines the root, or roots, of a polynomial. EX2/
EX3/ find the roots of f(x) = 4x2+ 10x-8, type: When the roots of a polynomial are known, the poly command can be used for determining the coefficients of the polynomial. The form of the poly command is: Ex4/ the coefficients of the polynomial in Ex1 can be obtained from the roots of the polynomial by:
Addition, Multiplication, and Division of Polynomials Addition: Two polynomials can be added (or subtracted) by adding (subtracting) the vectors of the coefficients. If the polynomials are not of the same order (which means that the vectors of the coefficients are not of the same length), the shorter vector has to be modified to be of the same length as the longer vector by adding zeros (called padding) in front. For example, the polynomials added by:
Multiplication: Two polynomials can be multiplied using the MATLAB built-in function conv, which has the form: The two polynomials do not have to be of the same order. Multiplication of three or more polynomials is done by using the conv function repeatedly. For example, multiplication of the polynomials f1 (x) and f2(x) above gives: which means that the answer is:
Division: A polynomial can be divided by another polynomial with the MATLAB built-in function deconv, which has the form:
